The strength of Hamilton's principle is obvious to me and I see the advantage. Now, for conservative systems we also have Maupertuis' principle that says:
$$ \delta \int p dq =0$$
and I am not sure how to derive an equation of motion from this? Is this of any use in practical computations? So, can one apply this principle for example to the harmonic oscillator?- I have never seen anybody using it.
Further, I read in Goldstein's classical Mechanics that the variation in Maupertuis' principle is not the one in Hamilton's principle, since we have constant Hamiltonian and changing time, whereas Hamilton's principle has constant time and varying Hamiltonian (in general).
I am a little bit wondering about this, since you could easily get Maupertuis' principle from Hamilton's principle: $$ \delta \int L dt = \delta \int p \dot{q} - H dt = \delta \int p \dot{q} dt = \delta \int p dq =0,$$ if $H$ is constant. Can anybody here explain to me, why we have to use a different variation and how one can use this principle?